Chapter 4
Laser Cooling of Alkali Atoms
Laser cooling [4] has become a very important tool of atomic physics research,
a fact that was recognised by the awarding of the Nobel prize in 1997 for its
discovery and advancement [1, 2, 3]. The techniques of optical molasses and
magneto–optical trapping have allowed ultra–cold temperatures of µKs to be
reached with arguably much greater ease than the cryogenic methods used by
the hydrogen community and other fields. It has opened the way for almost
routine production of quantum degenerate gasses [10, 11, 12, 66]. The fields
of atomic spectroscopy and atom clocks have surged ahead due to hugely reduced Doppler shifts and very long interaction times [9]. Laser cooling also
allows excellent access to the cold atomic sample for further manipulation or
interrogation [48].
In this chapter the theoretical principles and experimental implementation of
laser cooling will be briefly examined. The experimental methods to obtain
clouds of > 107 Rb atoms at 20 µK in 3 s will be discussed as will the effect of
different operating parameters on the loading of the MOT from a background
vapour of thermal atoms.
Two of the Nobel lectures from 1997 begin with “in 1978” [1, 3]. Since the
experimental field is then as old as I am (also, laser cooling of ions was first
demonstrated in 1978 [67, 68]) only a brief discussion will be taken.
34
Chapter 4. Laser Cooling of Alkali Atoms
4.1
35
General Introduction
In order to localize an object to a point a position–dependent force is required,
for example, Fx ∝ −x. Similarly, to reduce the velocity spread of a group
of atoms - i.e., to trap them in momentum space - a velocity dependent force
is needed; Fv ∝ −v. In laser cooling these forces arise from the interaction
Figure 4.1: An example of a position dependent force.
of an atomic ensemble with near resonant laser light. The laser linewidth is
typically stabilised to less than that of the linewidth of the atomic transition,
Γ = (2π)6 MHz. The laser detunings are typically < 3Γ, considerable less the
hyperfine splittings of the excited atomic states, which are typically on the order
of 100 MHz. The atomic system can then be well approximated by a two level
system.
Each absorption of a photon results in the atom acquiring a recoil velocity
of vrecoil = ~k/M (M = mass of atom) over the time of the transition to an
excited atomic state. The atom decays back to the ground state by spontaneous
emission of a photon of energy ~k0 , where k0 is the wave number corresponding
to a resonant photon, in a random direction. The time averaged impulse on an
atom due to many such spontaneous emissions is zero. The force on an atom
due to a single laser beam is then given by the product of the photon momentum
and the rate of photon absorption,
Fscatt = ~k ·
I/Isat
Γ
,
2 1 + I/Isat + (2∆/Γ)2
(4.1)
where I is the total intensity of light at the atom, Isat is the saturation intensity,
Γ = 1/τ is the linewidth of the excited state, τ is the lifetime of the excited
state and ∆ = ω − ω0 where ω0 is the resonant frequency of the EM transition
between the two atomic levels.
Chapter 4. Laser Cooling of Alkali Atoms
4.1.1
36
Doppler Cooling
Doppler cooling, as proposed in 1975 [69], provides frictional cooling for atoms,
i.e. a velocity dependent force. Doppler cooling makes use of the atomic velocity distributions and the narrow linewidths and frequency tuning available with
lasers. It can be assumed that if the force does go as F ∝ −v then it should be
possible to reduce the velocity of atoms down to zero. However, the stochastic
nature of the cooling mechanism means that atoms undergo a random walk,
which acts as heating. The threshold of Doppler cooling comes about by balancing of the cooling and heating through the spontaneous emissions, which
gives the limiting Doppler temperature of [4, 70],
TD =
For the D2 line in
85
Rb and
87
~Γ
.
2 kB
(4.2)
Rb, TD = 145 µK, and is the same order of
magnitude for the other alkalis [4, 71].
4.1.2
Sub–Doppler Cooling
Soon after the observation of temperatures close to the Doppler limit [14] temperatures six times lower than TD were observed in Na [72]. It was after this
experimental observation that it was realised that the multi–level nature of
atoms could allow cooling to much lower levels [73, 74]. Spatial variations in
the polarisations of the combined laser fields cause spatially varying light–shifts
of the magnetic sub–levels. The additional frictional force due to the presence
of the wells, along with spontaneous emission mean it is possible to cool within
these wells, a mechanism that was called Sisyphus cooling [73]. The theories
predicted that the minimum attainable temperature is limited by the depth of
the potential wells caused by these light–shifts,
Tmin
Ω2
∝
∆
I
∝
,
∆
(4.3)
(4.4)
where Ω is the Rabi frequency and
Ω=Γ
r
I
.
2 Isat
(4.5)
37
Chapter 4. Laser Cooling of Alkali Atoms
Again, following from Eqn. 4.4 we could presume that for I → 0 or ∆ → ∞
then Tmin → 0. This presumption fails when the energy gained from a photon
recoil is equivalent to the depth of the spatial potentials. A further reason is
that in the µK regime the atomic de Broglie wavelength,
λdB =
h
p
≈√
h
,
2 M kB T
(4.6)
becomes comparable to the wavelength of the cooling light and the extent of
the potentials. It then is not possible to localise the atomic wavepacket to the
potential wells. This new minimum temperature, the recoil limit is,
~2 k 2
M
= 2~ ω0
kB Trecoil =
= 2 Erecoil .
(4.7)
The experimentally realised minimum temperatures are approximately 10 Trecoil .
4.1.3
Magneto–Optical Traps
Optical molasses [75] provide cooling in momentum space but do not provide the
position dependent force to spatially trap atoms. Such a tool is the magneto–
optical trap (MOT), suggested by Dalibard and experimentally realised by Raab
et al. in 1987 [76]. The essential operation is most clearly seen in a 1D scheme
which can be extended to 3D. Here we consider transitions F = 0 → F 0 = 1,
but the technique is applicable to any transitions of the form F → F 0 = F + 1.
A linear magnetic field with a zero at z = 0 creates the energy level shifts shown
in Fig. 4.2. We take the B–field as the quantisation axis, allowing this axis to
change direction about z = 0. The Zeeman shift of a transition energy is
∆z = (gF 0 mF 0 − gF mF )
=
µ0 |B|
,
h
µB |B|
h
(4.8)
where gF , mF are the Landé g–factor and magnetic quantum number, µB is the
Bohr magneton and µ0 has absorbed the magnetic factors.
Chapter 4. Laser Cooling of Alkali Atoms
38
Figure 4.2: A 1D model for the operation of a MOT on a F = 0 → F = 1
transition.
We now introduce a pair of counter–propagating, red–detuned laser beams.
Both lasers are right–hand circularly polarized. A laser directed towards z = 0,
i.e., against the quantisation axis, drives σ − transitions, mF → mF 0 = mF − 1
in the atoms. A laser in the direction pointing away from z = 0, i.e., with the
quantisation axis, excites σ + transitions, mF → mF 0 = mF + 1. Because of the
Zeeman shift the mF = −1 level is always closest to resonance for the B–fields
we use. The ‘inward’ propagating beam, exciting σ − transitions, is scattered
more than the σ + beam and the atom is always being pushed towards z = 0.
From this model, it should be clear that each individual laser beam is both σ −
and σ + .
Raab et al. show that the force in the MOT is given by [15]
Fscatt = −α v − κ z ,
(4.9)
which provides the desired position and velocity dependent forces. The frep
quency for such a trap is κ/M , whereas the dampening rate is α/M . The
Chapter 4. Laser Cooling of Alkali Atoms
39
capture range for the MOT is limited by the requirement that ∆z < ∆.
The MOT can be extending into 3D using a laser arrangement of three, orthogonal sets of counter–propagating beams and a quadrupole field generated by a
pair of coils running in an anti–Helmholtz type configuration.
4.2
Loading of a Magneto–Optical Trap
The cliche of low temperature AMO physics must be “the MOT is the workhorse
of atomic physics”. Like Boxer in Animal Farm [77] the MOT, when setup,
works in almost all conditions and tolerates many of the errors of the experimentalist/farmer/pigs. To optimise the loading rate and number of atoms
trapped the alignment, polarisations and intensities of the laser beams, the
magnetic field gradient and the Rb vapour pressure must be set exactly, but
the MOT still works if any, or all of these is slightly wrong. However, in this
section the correct operation of the MOT will be discussed.
4.2.1
Experimental Setup
To create a MOT a good vacuum (Chapter 3), stabilised lasers (Chapter 2) and
a source of the element to be trapped (Section 4.3) are required.
The laser setup, prior to the experimental chamber is shown in Fig. 4.3. The
output beam from the trap laser has e−2 radii of 0.7 mm vertically and 1.4 mm
horizontally. The beam is passed through an expanding anamorphic prism
pair to give a beam which is has a circular cross section (±5%). The beam is
passed through an optical isolator to prevent undesired optical feedback into
the laser. A reference beam is split off from the trap laser on a polarising beam
splitter (PBS). This low power beam, typically 1 mW, is double passed through
a 80 MHz acusto–optic modulator (AOM) before being used for polarisation
spectroscopy, as described in Section 2.4. This means that the laser light output
from the laser is −160 MHz detuned from the lock frequency. The main beam is
then double passed through another AOM, the frequency of which can be varied
between 55 MHz to 100 MHz. This AOM thus allows changing of the cooling
laser beam frequency between −50 MHz and +40 MHz of the lock point.
Chapter 4. Laser Cooling of Alkali Atoms
40
Figure 4.3: Setup of the trapping/cooling laser and the repumping laser.
The hyperfine repumping laser has a more simple setup than the trapping laser.
The beam is similarly conditioned by passing through an anamorphic prism pair
and optical isolator. Approximately 1 mW of power is picked off by a thick glass
slide to provide two beams - pump and probe - for polarisation spectroscopy.
Chapter 4. Laser Cooling of Alkali Atoms
41
Due to the form of the polarisation spectroscopy spectrum the repumper is
locked to the F = 2 → F 0 = 1; 2 crossover. The main beam is sent single–pass
through an AOM to shift the frequency by the 78.1 MHz required to have the
laser resonant with the F = 2 → F 0 = 3 transition. This AOM also allows
fast switching of the laser power. A shutter was added after the AOM as up to
600 µW was found to ‘leak’ into the first order beam when the driver was set
to zero. The increased extinction of the repumper allows for increased loading
of the optical dipole trap and reduced trap loss [78].
The cooling and repumper lasers were combined on a PBS. The beams were
then expanded with a 5:1 telescope. Using a setup of λ/2 waveplates and PBSs
(Fig. 4.3) three beams with equal intensities of cooling light were obtained.
Due to the orthogonal polarisations of the cooling and repumping lasers unequal amounts of repumper pumper went into each beam. This effect was due
to combining the beams on a PBS but was not expected to affect the MOT
operation.
The MOT is formed from three orthogonal laser beams, which are retro–
reflected. The cooling beams were measured to have a radius of 7.5 mm (1/e2 )
before the chamber and a peak intensity of 9 (±0.2) mW cm−2 . Two of the
beams are in a vertical plane, each at 45◦ to the vertical, and one in the horizontal plane. Prior to entering the chamber each beam is passed through a
λ/4 plate to create the circular polarisation required of the cooling beams. On
exiting the chamber the beams are passed through another λ/4 waveplate and
then retro–reflected along the beam path to create the light field configuration
described in the previous section. For alignment purposes a variable aperture is
centred on each beam before the chamber. Using small beams allows for precise
alignment of the beams on the magnetic field zero. The retro–reflected beams
were aligned by aligning the returning beam on the back of the aperture.
Acousto–Optic Modulators (AOMS)
Three AOMs were used in the experimental setup. These are used to provide
frequency shifts of the laser beams and also to control the laser power to the
experiment in the cooling and repumping lasers. Concerns were raised by others
within the Durham AtMol group about the stability of the Isle Optics (TM)
Chapter 4. Laser Cooling of Alkali Atoms
42
Figure 4.4: Setup of the MOT lasers and magnetic coils. The direction of the
currents in the coils and the handedness of the polarisation of the beams are
indicated.
AOMs and drivers. However, the power to the experiment was measured to be
constant to better than 1% and was not observed to require time to stabilise,
apart from rise/fall time, after switching from on to off, or vice versa. The
frequency shift of the laser was more difficult to measure but the frequency from
the driver was monitored over 30 minutes after a frequency shift of 10 MHz,
which is typical of the shifts that are used during an experiment. The shifts
are measured with a frequency counter and hence could not be measured on
timescales shorter than required to take a reading. The results, shown in Fig. 4.5
show that there is a drift in the frequency over a time scale of 10 minutes, but on
a scale of only 10 kHz which is much smaller than the estimated laser linewidth
of 1 MHz and thus insignificant.
Chapter 4. Laser Cooling of Alkali Atoms
43
Figure 4.5: Frequency output of the AOM driver measured over 30 minutes
after a frequency shift of 10 MHz. The errors indicated are reading errors.
4.3
Atomic Source – Rb Dispensers
The source of Rb in the experiments are two SAES getter-dispensers, located in
the chamber as shown in Fig. 4.6. The active region of the dispenser is 1.2 cm
long and contains a compound of rubidium chromate along with a reducing
agent, see Fig. 4.7. Rb is released in a chemical reaction when the dispensers
are heated to the threshold reaction temperature of a few hundred ◦ C. The
exact threshold temperature of this reaction hasn’t been found in the literature
or from SAES themselves, but has been estimated as ∼ 500 ◦ C. This heating is
usually provided by passing a current to Ohmically heat the dispensers. The
dispensers are mounted on a Macor base to thermally and electrically isolate
from the vacuum chamber. They are connected in series and connected to
the ‘outside’ through 10 A vacuum feed–throughs. Kapton-coated copper wire
provides all connections within the vacuum chamber.
When first installed the dispensers must be run in. Existing literature [79, 80]
prescribes slowly increasing the current to heat the dispensers and evaporate off
contaminants while ensuring the pressure within the chamber does not increase
by more than an order of magnitude. We performed this procedure while baking
our chamber, see Section 3.5. The pressure and composition of gasses in the
chamber were carefully monitored with an ion–gauge and residual gas analyser
Chapter 4. Laser Cooling of Alkali Atoms
44
Figure 4.6: Photograph showing the position of the dispensers in the vacuum
system. The dispensers are the silver strips in the lower left of the large viewport. The wires over the dispensers are to prevent a direct line of light from
the thermal source to the MOT.
(RGA) as the current was increased up to 4.5 A. The pressure was not allowed
to rise above ∼ 10−6 Torr. The process took almost 24 hours.
During typically operation of experiments the dispensers were turned on at
2.5 A first thing in the morning. Over 30 minutes the steady–state number of
atoms captured in the MOT increased to a final value of 5 × 107 . The current
was then reduced to 2.3 A. The steady–state number captured in a MOT was
not observed to decrease. The current to the dispensers was turned off at the
end of each day. The ion gauge registered an increase of ∼ 0.2×10−10 Torr, from
Chapter 4. Laser Cooling of Alkali Atoms
45
Figure 4.7: Left) schematic of the SAES dispensers used in the experiment. The
metal case, contacts and sealing bar are made from stainless steel. Right) cross
section through a dispenser, showing the active material, which is a mixture of
rubidium chromate with a reducing agent.
1.2 × 10−10 Torr, when the dispensers were initially turned on, but recovered
within a minute.
Whenever there was a vacuum break we installed new dispensers, in the hope
that the chamber would never have to be opened again. After these vacuum
breaks running in was performed more quickly by increasing the current in steps
of approximately 0.2 A and then allowing the pressure to recover. This procedure took about 4 hours. There was no observable difference in the dispenser
operation after using this method.
4.4
MOT Magnetic Field Coils
To provide a magnetic field with a zero in the centre and a linear gradient
through the central region two coils were arranged in an anti–Helmholtz type
configuration. These MOT coils were formed from hollow wire with an external
square cross section of length 4.25 mm and a circular inner cross section of
diameter 2.25 mm. The large Ohmic cross–section lowers the resistance of the
coil, while the large surface area of the bore allows for cooling of the coils.
Chapter 4. Laser Cooling of Alkali Atoms
46
Cooling is provided by water, chilled to 14 ◦ C, pumped through the inner bore
of the coils. To protect against failure of water supply leading to the coils overheating a flow switch is positioned after the MOT coils. Tripping of this switch,
due to the flow rate of the cooling water going below a set value, shuts off current
to the MOT coils. The coils were wound on a mount made of Perspex made to
fit over the large windows on the vacuum chamber, Fig. 3.1. The coils were of
inner radius 75 mm, had 7 turns and were separated by 78 mm. The gradient of
the B-field through the centre was measured to be 0.0816 ± 0.002 G cm−1 A−1 .
The current to the coils is supplied by a Hewlett Packard 6671A power supply
Figure 4.8: B–field along the axis between the MOT magnetic coils for a current
of 150 A. The gradient in the centre of the coil is 12.33± 0.22 G cm−1 . The
thick line is an aid to show that this linear region is ∼ 2 cm on either side of
the centre.
capable of providing 150 A, giving a maximum B–field gradient of 12.25 G/cm.
A plot of magnetic field along the axis of the coils is shown in Fig. 4.8. Switching
of the current is with a bank of three MOSFETs in parallel with gates linked.
The MOSFETs are used as a digital switch. The drain of the MOSFET bank
was connected to the power supply and the source to the MOT coils, see Fig. 4.9.
A Schottky diode (International Rectifier 203CNQ100R) was placed across the
MOT coils to prevent oscillations in the MOT coil current during switching.
47
Chapter 4. Laser Cooling of Alkali Atoms
Figure 4.9: The MOT coil electronic setup. The resistor on the MOSFET gate
ensures the gate draws a current to fully open. The diode across the MOT
coils prevents voltage spikes during switching of the inductive load which could
damage the MOSFET.
Without a diode the coils can be switched off completely in a timescale of
100 µs but the current direction changes sign a number of times. The diode
slows the switching time down to 2 ms (from constant current to 0 A) but
provides a smooth, almost exponential decay of the current in the coils, as
shown in Fig. 4.10.
4.5
Loading Rates into the MOT
The number of atoms trapped in a MOT is a balance between the rate at which
atoms are captured R and the loss rate from the trap. The number of trapped
atoms over time is given by Monroe et al. as [81],
t
N (t) = Ns (1 − e− τ ) ,
(4.10)
where Ns is the steady state number of atoms and 1/τ is the loss rate due to
background collisions and is assumed to be independent of N .
In the experiment the partial pressure of the thermal vapour of Rb atoms is
determined by the current through the dispensers. Fig. 4.11 shows the variation
in the number of atoms in the MOT with time for increasing dispenser current.
We see that the trapped atom number is negligible up to a threshold current
48
Chapter 4. Laser Cooling of Alkali Atoms
120
100
Current (A)
80
60
40
20
0
−20
0
100
200
Timebase (µs)
300
0
1
2
Timebase (ms)
3
Figure 4.10: Effect of a diode across the MOT coils during switching off of the
current. Without the diode (bottom line), the current switches off completely
in 200 µs, but shows transient oscillations that could cause heating of trapped
atoms or could damage the MOSFET switches. With the diode (upper line)
the switch off takes 2 ms, but with a much smother turn off.
of approximately 3 A. The steady state atom number and loading rate then
increase rapidly up to a dispenser current of ∼ 4.25 A when the steady state
number begins to saturate. At the highest currents, corresponding to the highest
background pressure of Rb, the steady state atom number peaks due to the
increased trap losses from collisions with hot thermal atoms. The loading rate
of atoms is shown explicitly in Fig. 4.12.
4.6
Optical Molasses
To achieve good molasses cooling the intensities of the laser beams need to be
balanced and the magnetic field in the molasses region has to be reduced to zero.
The intensities of the molasses beams were measured before the chamber and
were matched to better than 5%. The Earth’s B–field cancelling coils allowed
bias fields of up to 0.6 G to be applied in orthogonal directions with a resolution
Chapter 4. Laser Cooling of Alkali Atoms
49
Figure 4.11: The variation of atom number in the MOT with time for varying
heating currents through the dispensers. For larger currents the steady atom
number reaches a maximum and then begins to decrease at the highest currents
as a result of increased losses due to the increased background pressure.
of 5 mG. The cancelling coils were optimised by reducing the B–field gradient
of the MOT coils and adjusting the cancelling coils until the MOT position was
observed to not move with varying MOT B–field gradient. This indicates that
the MOT centre corresponds to a B–field zero. In practice the individual fields
were adjusted until the cold atom cloud showed slow, isotropic expansion when
the MOT magnetic fields were suddenly switched off while leaving the lasers
fields on.
It was found that a molasses duration of ≥ 10 ms was required to achieve the
lowest temperatures in the optical molasses. For shorter and longer durations,
the shot–to–shot temperature was found to vary up up to ±10%, Fig. 4.13.
Chapter 4. Laser Cooling of Alkali Atoms
50
Figure 4.12: Loading time, τ (Eqn. 4.10) of the MOT as a function of dispenser
current.
4.7
4.7.1
MOT Diagnostics
Atom Number
A knowledge of the number of atoms trapped is desired. Also, the fill rate of
the MOT is a good diagnostic of the MOT operation and optimisation. Both
these properties, can be measured by monitoring the scattered fluorescence
from the MOT region over time with a sensitive photodiode. The total power
measured at the photodiode will be give by the product of the energy per
scattered photon, the solid angle over which photons are measured, the number
of photons measured and the rate photons are scattered,
Ppd =
hc Ω
·
· N · Γsc .
λ 4π
(4.11)
The fractional solid angle can be approximated by
Ω
π r2
≈
,
4π
4 π R2
(4.12)
Chapter 4. Laser Cooling of Alkali Atoms
51
Figure 4.13: Variation of temperature with molasses duration. Data were taken
for 107 atoms at a molasses detuning of ∆ = −5 Γ and single beam intensities
of 4 mW/cm2 . Note that the temperature before molasses was measured as
190 µK.
where r is the radius of the lens collecting the scattered light and R is the
distance from the MOT to the lens. By using a large area lens close to the
MOT and focussing the collected light on the photodiode the percentage of
scattered light collected will be greatly increased. The scattering rate can be
calculated from
Γsc =
I/Isat
Γ
.
2 1 + I/Isat + (2Γ/∆)2
(4.13)
In this equation it is unwise to use the value of Isat = 1.6 mW/cm2 , the saturation intensity for the closed mF = 3 → m0F = 4 transition. During imagining
all polarisations will be excited and Isat will be larger for all transitions other
than the closed transition. The guide, BEC For Everyone [66] suggests using a
value of Isat = 4.1 mW/cm2 .
In the experiment a f = 10 cm lens, with an open aperture of 4.8 cm, 20 cm
from the MOT was used to collect light scattered by the cold atoms. For
Chapter 4. Laser Cooling of Alkali Atoms
52
Figure 4.14: Photodiode circuit used to collect scattered fluorescence. RG =
1 MΩ converts the current signal from the photodiode into a voltage. The two
switches allow for four levels of gain.
MOT parameters of 9.2 mW/cm2 per beam at a detuning of −2 Γ then Γsc =
(2π)1.2 MHz. The power estimated at the photodiode, from Eqn. 4.12, is 6.7 ×
10−15 W/atom. For enhanced signal–to–noise, the photodiode was mounted
in a light proof box with a narrow band filter which was measured to transmit
only 65 % at 780 nm. The photodiode used, (IPL10050 CW) has a sensitivity of
0.45 A/W which then corresponds to a signal of 1.9× 10−15 A/atom. Using a two
stage gain of 106 , Fig. 4.14, we have a measurable signal of 5.3 × 10−6 atoms/V.
To measure larger clouds the gain after the photodiode stage can be reduced
by ×10.
4.7.2
Temperature
Due to the magnetic field gradient of the MOT coils being a factor of two higher
in one direction the cloud has a pancake shape with a ratio of the widths of
2:1. Technical issues, such as the laser beams being not perfectly Gaussian and
noise in the laser frequency and intensity, mean that the cold atom cloud has an
irregular shape. Arnold et al. indicate that the temperature and density vary
across a MOT but do indicate that a Gaussian density distribution is valid [82].
The temperature of the cloud of atoms was measured using a time–of–flight
(TOF) technique, as introduced by Lett et al. [75]. The atomic cloud is released from the trapping fields and the spatial distribution is monitored over
time. By comparing this distribution to a Maxwell–Boltzmann distribution the
Chapter 4. Laser Cooling of Alkali Atoms
53
Figure 4.15: Set up of the imaging optics.
temperature can be found [83].
We assume that density of atoms has a Gaussian dependence,
2
1
− x2
2σ
g(x) = p
e x ,
2πσx2
(4.14)
and similarly for velocity, assuming a Maxwell–Boltzmann distribution
r
m v2
m
− 2 k xT
f (vx ) =
e B x
2π kB Tx
r
m(xf −xi )2
m
−
e 2 kB Tx t2 ,
(4.15)
=
2π kB Tx
where we make use of
xf = x i + v x t .
(4.16)
The final position distribution is also a Gaussian and can then be found by a
convolution of Eqns. 4.14 and 4.15. The expansion of the MOT cloud is then
of the form,
r
k B Tx 2
t .
(4.17)
m
By measuring σx , the Gaussian width of the cloud along the x (or equivalently
σx (t) =
σx2 (0) +
the y or z directions) for different release times, the temperature is
m σx2 f − σx2 i
.
Tx =
kB t2f − t2i
(4.18)
54
Chapter 4. Laser Cooling of Alkali Atoms
Alternately, plotting t2 against σx2 gives a graph of slope Tx · kB /m and intercept
σx2 (0) from which the temperature is easily extracted.
Experimentally we observe that the observed time–of–flight data follows the behaviour predicted above. We have not used any other methods for temperature
measurement but as the method described here is the standard method in the
cold atom community we have some faith in the results.
4.8
Laser Heating of Dispensers
In December 2004 electrical continuity across the vacuum feed–throughs was
found to have been lost; in other words the resistance across the feed–throughs
went from 0.4 Ω to open circuit with no warning or sign. No cause could be
discovered and all connections appeared, to the eye, to be okay.
It is not important to the emission of a Rb vapour how the dispenser is heated, as
long as the active material is heated above the threshold temperature. The experiment has a 10 W, Lightwave Electronics 220 Nd:YAG laser (λ = 1.064 µm)
as part of the setup that was used in the previous generation dipole trapping
experiments [84]. It was attempted to use this laser to heat the dispensers.
4.8.1
Coupling of Laser Radiation to Metals
The theory of coupling energy from a laser beam to a metal has been mainly
examined for laser cutting applications, where a focussed laser beam moves
relative to a metal surface and melts the material [85, 86]. The situation looked
at here is for heating of a metal bar in thermal contact with thin sheets of the
same metal and also with a complex chemical compound. We will only look
briefly at the theory as a detailed examination would be time consuming and
quite possibly fruitless. The question that we wish to answer is whether we can
heat a piece of stainless steel to > 500◦ C with less than 10 W of Nd:YAG laser
power. To do this we need to know about the thermal and optical properties of
stainless steel and the coupling of laser radiation at 1.064 µm to stainless steel.
The heat flow problem for a homogenous and isotropic solid is [87]
∇2 T −
A
1∂T
=− ,
κ ∂t
K
(4.19)
55
Chapter 4. Laser Cooling of Alkali Atoms
where T is the temperature, A the rate at which heat is supplied per unit time
per unit volume, K is the thermal conductivity and κ is the thermal diffusivity.
κ is given by other known quantities as
κ=
K
,
ρC
(4.20)
with ρ the material density and C the heat capacity. For a CW Gaussian beam
of power P focussed to a 1/e2 radius w0 Bass gives a solution for the temperature
at the laser centre on the surface as [87],
√
1/2
8κt
4 2αP
−1
tan
T (t) =
.
K π 3/2 w0
w02
This clearly implies a limiting temperature of
√
2 2αP
.
T |t→ ∞ →
K π 1/2 w0
(4.21)
(4.22)
Using the data provided by Bass [87] for stainless steel,
K = 0.26
(W/cm ◦ C)
C = 0.6
(J/g ◦ C)
ρ
= 8
(g/cm3 )
κ
= 0.054 (cm2 /s)
α = ∼ 0.1
implies, for a power of 2.5 W, a limiting temperature of 440◦ C. However it also
implies that this temperature is reached very quickly, on the order of 10 ms.
However this assumption is for a solid, isotropic amount of stainless steel. The
dispenser has a thin wall of steel and is then in contact with the chemicals
within. The thermal conductivity and correspondingly, the thermal diffusivity
are sure to be lower (no data is available) so it can be postulated that the
maximum temperature achieved will be higher and the rate of temperature
increase slower than predicted for solid stainless steel due to the lower values of
κ and K.
4.8.2
Laser Induced Emission of Rb from a Dispenser
The examination above would appear to give lower estimates of the limiting
temperatures and time taken to achieve these. Even with these considerations
56
Chapter 4. Laser Cooling of Alkali Atoms
we still had no feeling of the result that could be obtained, so we took a conservative, systematic approach. The Nd:YAG laser was passed through a λ/2
waveplate and a PBS, in order to control the power. The laser was then focussed onto the dispenser, through the large viewports, with a f = 10 cm lens,
Fig. 4.16. The peak intensity at the laser focus is given by,
I0 =
2P
.
π ω02
(4.23)
Due to the construction and orientation of the dispensers the Nd:YAG beam
had to be focussed at an angle through the 6 mm thick viewports. This caused
significant astigmatism of the beam, with the vertical and horizontal foci being
separated by 7.8 mm. The minimum spot sizes in each direction were 21 µm
vertically and 24 µm horizontally. The minimum area of the focus, the ‘circle of
least confusion’ had a 1/e2 radius of 35 µm. From Eqn. 4.23 the peak intensity
at the focus is then, I0 = 5.2 × 104 W/cm2 per W of laser power. A CCD
camera, with a 780 nm filter to block out scattered λ = 1.064 µm light, was
positioned to monitor the position the Nd:YAG laser was focussed to and also
to estimate the efficiency of the heating due to the thermal glow, Fig. 4.17. The
fluorescence from the MOT was recorded to monitor the efficiency of the Rb
emission. It was quickly found that if the laser was focussed with a power of
Figure 4.16: Schematic of focussing of the Nd:YAG beam onto a dispenser. The
intense laser beam produces a thermal atom source that loads a MOT in the
centre of the chamber. A magnified view of the dispensers is shown inset.
Chapter 4. Laser Cooling of Alkali Atoms
57
Figure 4.17: CCD images of alkali dispensers without (left) and with (right)
the heating beam. The beam spot lies on the bar above the active region, see
Fig. 4.7.
1 W on the bar that protects the active material within the dispenser then the
MOT fluorescence was observed to increase. Increasing the laser power to 2 W
showed a rapid increase in the MOT fluorescence.
Experiments were conducted to examine the potential of the technique for our
loading experiments [88]. Loading of the MOT was performed by turning on the
dispensing laser for 5 s and turning on the MOT at a variable delay relative to
the end of the heating pulse. Fig. 4.18 shows the collected fluorescence from the
MOT for delays of a) +10 ms, b) −2.5 s and c) −5.0 s. A dramatic reduction in
collected fluorescence was observed when the MOT was turned on 10 ms after
the dispensing laser was extinguished. Based on the signal–to–noise ratio of the
photodiode signal we would place a conservative upper limit of 100 ms on the
switch off time of the atom source.
Increasing the dispensing beam power increased the number of atoms trapped.
For example, for a pulse time of 2.7 s, increasing the power from 2 W to 3 W
increased the number of trapped atoms by a factor of 2.6 to 4.2×107 . Further
increasing the power to 4 W increased the number of trapped atoms to 7.5×107 .
However, at higher powers the recovery time of the background pressure also
increased to many seconds, measured by increased loss from the optical dipole
trap. Increasing the pulse duration results in larger trapped atom numbers, as
should be expected from Fig. 4.18. The equilibrium trapped atom number, for
laser powers ≥ 2 W was measured as > 109 , limited by the MOT laser beam
size. This equilibrium number took over 100 s to achieve and the background
Chapter 4. Laser Cooling of Alkali Atoms
58
Figure 4.18: Atomic fluorescence signal from the MOT for different times between turning the dispensing laser off and the MOT field on. In each case 2 W
of the dispensing laser is pulsed on for 5 s, and turned off at t=0 s. The MOT
B field is on at (a) t=0.01 s, i.e., 10 ms after the dispensing pulse is switched
off, (b) t=-2.5 s, and (c) t=-5 s, i.e., MOT on at the start of the dispensing
pulse. Line (a) shows that the switch off time of the Rb flux is on the order of
10 ms.
pressure took many minutes to recover.
The optimum position of the heating laser focus was along the bar which protects the active region within the dispenser, see Figs. 4.7, 4.16. The flux of atoms
varied with the position of the heating beam along the bar. Also, when the dispensing laser was used repeatedly on a single spot, over a period of weeks, the
number of atoms captured in the MOT began to decrease by 10 − 20%. However, by moving the focus by ∼ 100 µm along the bar the original flux was
recovered.
To obtain a measure of the repeatability of light induced dispensing, the MOT
was loaded on a cycle of fixed loading times. Fig. 4.19 shows the atom number
as a function of time for pulsing on both the dispensing laser and the MOT
trapping fields for 4.1 s, followed by 3.5 s of no dispensing or MOT trapping.
We typically loaded 2.4 × 107 atoms with a standard deviation of 106 . We
observed that using lower powers (0.5 W) increased the shot–to–shot variation
Chapter 4. Laser Cooling of Alkali Atoms
59
Figure 4.19: Trapped MOT atom number from repeated pulsing of the dispensing laser (2 W for 4 s) and MOT trapping fields. The mean of the peak number
of atoms loaded is 2.4 × 107 atoms, with a standard deviation of 106 .
of number of atoms trapped in the MOT.
It was observed that the switch–off time of the atom source was very rapid,
though the turn on time could be greater than 1 s. This can be explained
as follows: when the dispenser is heated by the dispensing–laser beam the
temperature locally rises until the laser is extinguished, assuming the laser
is not on long enough to reach a steady state temperature. When the laser
is extinguished the temperature falls. Fig. 4.20 illustrates the temperature
variation as a function of time. As the alkali metal vapor emission is a threshold
phenomenon the turn off time can be considerably faster than the turn on time.
However, if the laser is switched on again before the local temperature reaches
“room temperature” then the local temperature will reach the threshold value
more rapidly than before. In a pulsed sequence the time taken to reach the
threshold temperature will decrease and the local maximum temperature will
increase for each pulse. However, these effects saturate to steady state values
after a few cycles due to the balance between the heating and thermal losses.
This behaviour is confirmed by our experimental observations, namely when
the dispensing laser is used in single–shot experiments the time constant to
reach a threshold atom number is greater than that in pulsed experiments. In
60
Chapter 4. Laser Cooling of Alkali Atoms
Fig. 4.18, the time taken to start dispensing atoms is 1.3 s whereas in a pulsed
experiment, such as shown in Fig. 4.19, the time taken to start dispensing is
temperature
only 0.2 s.
Turn Off
Turn On
time
Figure 4.20: Schematic of the temperature vs. time behavior of the active region
of the dispenser. The figure shows the difference in the ‘turn on’ and ‘turn off’
times of the alkali metal vapor release for a single pulse of the heating laser.
The dashed, horizontal line marks the threshold temperature for emission of
the alkali metal vapor
A shutter was used along the Nd:YAG beam to switch the emission of the Rb
vapour. By triggering the shutter closing from the captured MOT fluorescence
we could stabilise the number of atoms loaded into the MOT. A Schmitt trigger was used, as it provides stable switching from a signal with noise, due to
hysteresis in the switching [89]. The simple circuit used is shown in Fig. 4.21.
The triggering voltages are
R1 ||R2 ||R3
R1 ||R2 ||R3
Vref +
Vref
R1
R3
R1 ||R2 ||R3
R1 ||R2 ||R3
Vref −
Vref ,
=
R1
R3
Vup =
Vdown
(4.24)
(4.25)
Chapter 4. Laser Cooling of Alkali Atoms
61
where R1 ||R2 ||R3 is the parallel resistance of the three resistors. Hysteresis
occurs in the switching due to the sign difference in Vup and Vdown . In the
circuit used, Fig. 4.21, Vref = 5 V.
Figure 4.21: Circuit diagram of the Schmitt trigger used to trigger the start of
the experiment from the fluorescence photodiode signal. The first inverter is
used as the photodiode circuit outputs a negative voltage. The Schmitt trigger
section also outputs a negative voltage which is also inverted. A circuit without
inverter stages was attempted but not found to work. The FET is opened when
the trigger is switched by the photodiode signal. The values of the resistors are;
R1 = 4.7 kΩ R2 = 200 kΩ, variable, R3 = 200 kΩ.
The use of high powered diode lasers was also examined as a cheaper laser
source. High power laser diodes (> 1 W) with high quality spatial mode are
available but cost–per–Watt are more expensive than many Nd:YAG lasers.
A cheaper option is to use a diode bar laser, though the poor beam quality
from this type of laser makes it difficult to achieve a small spot size and the
corresponding high intensity required. We used 2 W diode bar laser (High
Power Devices HPD1120 2 W bar) and observed a Rb flux from the dispensers.
However, the number of atoms captured, and the capture rate in the MOT were
an order of magnitude lower than from using 2 W of Nd:YAG power. This was
attributed to the better quality of the Nd:YAG beam.
As an aside, while the use of this technique was a discovery born out of desperation we have found that the technique was very useful for our experiments
and has some advantages over existing sources of atoms for laser cooling. The
main attributes of the laser induced emission are a high atom flux followed by
Chapter 4. Laser Cooling of Alkali Atoms
62
a very fast switch off time. This allows large clouds of atoms to be trapped
without compromising the background pressure or trap lifetimes. Nearly all
BEC experiments use a separate chamber or a large Zeeman slower to precool
the atoms from a background vapour and then transfer to a science chamber for
experiments. This increases the vacuum and laser requirements of the experiment and adds much complexity. A technique such as laser induced emission
of atoms could remove the need of a precooling stage.
4.9
Shutters
Figure 4.22: Typical switching characteristics of the shutters used in the experiments. The shutter is positioned at the focus of a telescope, with an estimated
beam waist of 20 µm, and the transmitted power is monitored on a photodiode.
The trigger pulse is shown in light grey and the transmitted light is shown in
black. The delays, ∆t1 and ∆t2 are constant to better than 50 µs.
Shutters, based on the Singer et al. design [90] were used on the repumping
laser and also on the Nd:YAG laser used to heat the dispensers. The shutters
use a modified speaker coil which has the speaker cone trimmed off, to reduce
weight. A lightweight ‘flag’, made from aluminium is moved to block the beam.
Chapter 4. Laser Cooling of Alkali Atoms
63
The coils are powered by a simple FET circuit which can then be triggered from
LabVIEW. Typical performance of the shutters is shown in Fig. 4.22.
The shutters compared well with expensive, commercial shutters used in other
experiments within the group in that they displayed very little jitter when either
opening or closing. The jitter was on the noise level of the oscilloscope, and
was estimated at < 50 µs. Over the course of a year 2 shutters failed, with no
sign of wear before ceasing to work completely. Replacing a shutter is a quick
and easy job and characterising the new shutter merely requires a photodiode
in the beam.